Compact microwave cavity for high performance rubidium
frequency standards
Camillo Stefanucci,1,a)Thejesh Bandi,2,a)Francesco Merli,1Matthieu Pellaton,2 Christoph Affolderbach,2Gaetano Mileti,2and Anja K. Skrivervik1
1Laboratoire d’Électromagnétisme et d’Acoustique (LEMA), École Polytechnique Fédérale de Lausanne,
CH-1015 Lausanne, Switzerland
2Laboratoire Temps-Fréquence (LTF), Institut de Physique, Université de Neuchâtel,
CH-2000 Neuchâtel, Switzerland
The design, realization, and characterization of a compact magnetron-type microwave cavity operat-ing with a TE011-like mode are presented. The resonator works at the rubidium hyperfine ground-state frequency (i.e., 6.835 GHz) by accommodating a glass cell of 25 mm diameter containing rubid-ium vapor. Its design analysis demonstrates the limitation of the loop-gap resonator lumped model when targeting such a large cell, thus numerical optimization was done to obtain the required per-formances. Microwave characterization of the realized prototype confirmed the expected working behavior. Double-resonance and Zeeman spectroscopy performed with this cavity indicated an excel-lent microwave magnetic field homogeneity: the performance validation of the cavity was done by achieving an excellent short-term clock stability as low as 2.4× 10−13τ−1/2. The achieved experi-mental results and the compact design make this resonator suitable for applications in portable atomic high-performance frequency standards for both terrestrial and space applications.
I. INTRODUCTION
Rubidium (Rb) atomic clocks1, 2 are used as secondary frequency standards. Their compactness and competitive stability—over one to several days—make them interesting and suitable candidates for satellite navigation and telecom-munication systems applications.3
Most of the commercial Rb atomic clocks are based on the double-resonance (DR) spectroscopy scheme illustrated in Fig. 1 that requires two electromagnetic (EM) fields: an opti-cal field at νopt= 384.23 THz for the Rb D2-line to polarize the atoms by optical pumping, and a microwave field at νRb (unperturbed value4equals to 6 834 682 610.90 429(9) Hz) to drive the ground-state hyperfine clock transition that serves as atomic frequency reference.
Cylindrical,5 rectangular,6 or mixed-design microwave cavities7 are usually used to apply the microwave field to the Rb atoms contained in vapor form inside a glass cell. In order to achieve a compact design fulfilling the electro-magnetic requirements for our application,8the loop-gap res-onator (LGR)9 was selected as a first model. This typology has a broad field of applications including electron spin reso-nance (ESR),10electron paramagnetic resonance (EPR),9and nuclear magnetic resonance (NMR)11as it provides excellent EM characteristics within a volume three times more compact than a simple cylindrical cavity.
The microwave magnetic field distribution of the LGR, also named split-ring or slotted-tube resonator,12, 13is interest-ing for frequency standards application where it is known as magnetron cavity.14 Examples in portable ground and space atomic clocks include the rubidium atomic frequency stan-a)C. Stefanucci and T. Bandi contributed equally to this work.
dard (RAFS), where the cavity holds a vapor cell of 14 mm diameter and resonates at 6.835 GHz,14, 15and the passive hy-drogen maser (PHM)16 accommodating a hydrogen storage bulb of∼60 mm diameter and resonating at 1.42 GHz.
This work focuses on a compact magnetron cavity to guarantee a clock signal with an improved short term stability better than the conventional RAFS and PHMs.17Recently, the stability of atomic vapor cell standards has been improved by the use of lasers for state preparation.1, 18–20 The short-term frequency stability of a clock (in terms of Allan deviation) is inversely proportional to the atomic quality factor and to the signal-to-noise (S/N) ratio.5 This implies that a narrow linewidth of the atomic clock signal increases the atomic qual-ity factor and thereby improves the short-term clock stabilqual-ity. A larger dimension of the rubidium cell results in a higher atomic quality factor and allows interrogating more atoms with the standing microwave field, and hence, gives a better S/N ratio. Previous studies were done on magnetron-type cav-ity that can accommodate 14 mm diameter cells, whereas here we develop and study a larger cavity that can hold an enlarged cell of 25 mm diameter.
To summarize, this paper focuses on the design of a res-onator for enlarged Rb cells while still ensuring the compact-ness of the overall system (LGR cavity).
The manuscript is organized as follows: Sec. II describes the LGR model and the design of the magnetron cavity which resonates around νRb in a TE011-like mode. Analytical and numerical investigations are carried out in order to evaluate the limitations of the lumped elements equivalent model and to compute the performance of the cavity. This is followed by the simulated and experimental microwave characterization of the resonator in Sec. III. The field mode analysis of the de-signed magnetron cavity in double-resonance spectroscopic
Rb Atoms
Laser (ν ) Microwave excitation (ν ) PhotodetectorPhotocurrent
ν
Rbν
z
C-field coil Rb optFIG. 1. DR spectroscopy principle: Rb atoms inside a cell are excited by a laser beam at νoptand an EM field at νRb. The clock frequency is determined
by a photodetector. The C-field is a dc magnetic field to separate the hyperfine energy levels by Zeeman splitting5and it identifies the quantization axis1
(it coincides with the z-axis).
experiments is presented in Sec. IV. Finally, Sec. V validates the cavity by demonstrating the frequency stability of our clock, while conclusions are drawn in Sec. VI.
II. MICROWAVE CAVITY DESIGN A. Loop-gap resonator model
The LGR consists in a metallic slotted loop inside a cylin-drical shield. The simplicity of the structure allows the use of a lumped elements equivalent model to estimate the resonance frequency10, 21νr, νr = 1 2π n π r2εμ t w LC 1+ r 2 R2− (r + w)2 shield 1 1+ 2.5t w fringing , (1) where r is the inner loop radius, R the shield radius, n the number of gaps of thickness t, and width w as illus-trated in Fig. 2(a). The three indicated terms identify the contributions of the basic inductor-capacitor (LC) equiva-lent circuit model, the correction due to the presence of the outer shield and the factor related to the magnetic fring-ing field. The correspondfring-ing resonant mode has the
elec-Electrodes Shield Supports x y z x y z Gap w t r R Cell
FIG. 2. Basic loop-gap resonator (a) and magnetron cross section (b) with n = 4 gaps. The metallic pieces and their separations are identified as electrodes and gaps, respectively. Dashed line identifies the glass cell.
tric field mainly concentrated in the gaps (parallel plate ca-pacitor), while the magnetic field is confined within the loop.22 This mode will be named here as TE011-like in analogy with cylindrical cavities, however, the transverse field distributions are different with respect to the circular geometry.
Equation (1) serves as an excellent base to design a LGR, but a more precise full-wave numerical analysis is required since the lumped model does not take into account the fol-lowing aspects:
(a) The dielectric cell which contains the atoms in the vapor form.
(b) The metallic supports which attach the electrodes to the outer shield as in a magnetron cross section,23Fig. 2(b). The two aspects imply frequency shifts >3%, which are not negligible in our application.
Additionally, two other problems must be taken into account:
(a) Equation (1) does not consider the effects of the open-ings of the cavity required to let the laser beam interact with the Rb atoms.
(b) Equation (1) is a single mode model and thus gives no information about other excited resonances in the same frequency range of the TE011-like mode.
These four issues will be further detailed in Subsections II A 1 and II A 2, highlighting the main differences between the design of small LGRs and enlarged cavities.
1. Dielectric cell and metallic supports
In order to appreciate the effect of the presence of the cell and supports, two significant examples are reported: one LGR of shield radius R= 10.2 mm (for a Rb cell of 14 mm diame-ter) and another one of R= 18.0 mm (for a 25 mm diameter Rb cell). Other parameters are set to obtain the targeted νRb (n= 4, 6; t = 0.841, 2.138 mm; w = 1.8, 3.8 mm; r = 7.5, 13.5 mm; R= 10.2, 18 mm, for the two cases, respectively).
The relative errors between the resonant frequencies of the TE011-like mode of real structures and the result of Eq. (1) are reported in Fig. 3. Four different geometries are numeri-cally investigated with theANSYShigh frequency simulation software (HFSS):24 basic LGR, basic LGR with a dielectric cell (εr= 4.5, thickness = 1 mm), basic LGR with supports,
basic LGR with supports and dielectric cell.
Two different effects can be distinguished: while the presence of cell reduces the resonant frequency, the sup-ports entail an upwards frequency shift. As a consequence the lumped model is shown to have an error around 10% (which can reach 25% for certain geometries). It can be noticed that the errors are more prominent for the larger cell.
2. Apertures and spurious modes
Figure 4 illustrates an LGR terminated by two open cir-cular waveguides. These boundary conditions decrease the
Basic LGR Basic LGR + cell
Basic LGR + supports Basic LGR + supports + cell Value from Eq. (1)
0 5 10 15 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 R = 18 mm Relative error [%] Freq u ency [GHz] 0 5 10 15 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 R = 10.2 mm Relative error [%] Freq u ency [GHz] 1.6%
FIG. 3. Relative error of the simulated24eigenmode frequency of the TE 011
-like mode with respect to the value calculated by (1) for four different geometries.
Q-factor depending on the apertures’ dimensions. To under-stand the importance of the latter aspect when the shield di-ameter is increased, let us refer to the work already published by Mett.25In particular, the two following aspects are impor-tant to be recalled:25
r
The central part can be modeled as a ridgedcircu-lar waveguide:25 the field distribution of the resonant frequency of LGR is as the TE011 mode of cylindri-cal resonators perturbed by the presence of the elec-trodes (becoming TE011-like). However, other trans-verse modes can be excited.
r
The top and bottom loading sections may support TEand TM transverse modes of a classic circular waveg-uide (thus, a different field distribution from the fring-ing configuration of the TE011-like mode).
FIG. 4. Magnetron cavity model as a ridged waveguide with two circular loading waveguides. Apertures are sketched in dashed lines. The shield is not reported for clarity reason.
A full-wave analysis helps us to understand the role of apertures when the single propagating structures are consid-ered. If a propagating mode is excited in the loading sections, it couples with the openings implying a modification of the field distribution in the central region and a stronger decrease of the Q-factor because of the related radiation. For this rea-son, the terminating waveguides are usually designed to work under cut-off.10This means for applications at νRbsetting the maximum radius of the empty circular waveguides equal to Rc= 12.85 mm.
In general, setting R > Rcrequires the precise analysis of the loading sections. Nonetheless, the previous conservative limit to the radius can be circumvented by paying a partic-ular attention to the supported modes in the terminating cir-cular waveguides. In fact, the TE01-like transverse mode of the ridged region can propagate in the loading sectors only if R > Rmax = 26.75 mm. Therefore, we can conclude that the resonant field of a LGR with an outer shield of radius Rc
< R < Rmaxcan still be insensitive to the apertures. Obviously, the latter is also true if other modes (such as the fundamental TE11) are not excited by the discontinuity between the ridged section and the loading waveguides.
The increase of R implies also the presence of more modes, which may prevent the correct functioning of the Rb atomic clock. Indeed, a field distribution with transverse mag-netic components Htdrives unwanted σ Zeeman transitions26 (see Sec. IV B).
Table I reports the relative frequency difference of the TE011-like mode and the closest resonant mode with consis-tent Ht. It is evident that for the larger cavity the modes are closer in frequency, and the combination of the described ef-fects can lead to undesirable configurations where both modes overlap (f < 5%). It is worth reminding that this situation can pose problem for tuning of the cavity and the correct func-tioning of the clock.
B. Magnetron cavity design
After a first estimation of the geometrical dimensions via Eq. (1), a magnetron cavity with six electrodes was designed. Iterative optimization procedure was performed to include all the details of the final realization into the numerical simula-tions. The proposed structure is depicted in Fig. 5. A coaxial cable loop placed under the middle of the electrodes excites TABLE I. Relative frequency difference f = (fTE− fHt)/fTEbetween
the simulated TE011-like mode (fTE) and the first Htmode (fHt)afor the same geometries of Fig. 3.
R= 10.2 mm Basic +Cell +Suppression +Cell + suppression
fTE(GHz) 6.845 6.496 7.532 7.062
fHt(GHz) 7.495 5.321 8.676 7.906
|f| 9.5% 17.7% 15.2% 11.9%
R= 18 mm Basic +Cell +Suppression +Cell + suppression
fTE(GHz) 6.610 6.333 7.782 7.329
fHt(GHz) 6.147 7.163 7.751 7.121
|f| 7.0% 13.1% 3.1% 2.84%
aNotice that f
FIG. 5. Computer-aided-design (CAD) of magnetron cavity with six elec-trodes and a coaxial excitation. The closing cap is not reported to appreciate the electrodes shape.
the cavity and ensures a good coupling27 with the desired TE011-like mode. The shield, R= 18 mm, and the electrodes allow the insertion of a glass cell of 25 mm in diameter. The latter can be precisely positioned with a metal cap inserted into the cavity; this mechanism allows also the fine tuning of the resonator at 6.835 GHz within a 200 MHz range (<3%).15
III. MICROWAVE CAVITY CHARACTERIZATION A. Simulated performances
The simulated microwave magnetic field distribution of the proposed magnetron cavity is reported in Fig. 6. The H-field distribution of the TE011-like mode is uniform along the z-direction with small radial and azimuthal components. It is important to notice that the difference between the resonance frequency predicted by (1) and the full wave numerical anal-ysis is larger than 25% including all the geometrical details of the final design.
To characterize the microwave magnetic field distribution for DR frequency standards, we considered the filling factor η and the field orientation factor ξ.
The filling factor—mainly used in the context of active hydrogen masers—is defined as5
η= 1 Vcell (V cellHzdV) 2 Vcavity|H| 2dV, (2)
where Vcell is the volume occupied by Rb atoms (excluding the reservoir) and Vcavity is the total volume of the cavity in-side the shield including the cell. The z-direction here is
sup-Lens Coaxial cable Return magnetic flux z
FIG. 6. Simulated H-field distribution at 6.835 GHz (side cut) with an in-put power of 1 W. The analysis includes the semi-rigid coaxial cable loop, the optical lens needed to focus the light onto a photodetector and the glass cell.
posed to coincide with the direction of the static magnetic C-field defining a quantization axis and it is parallel to the light propagation direction1 as shown in Fig. 1. Thus, η measures the fraction of magnetic field energy useful for the clock sig-nal (Hzdrives the ground-state clock transition as detailed in
Sec. IV) compared to the total microwave energy stored in the cavity. It is a measure of the efficiency of the cavity for cou-pling the overall microwave energy to the atoms stored in the cell, which is an important parameter for an active hydrogen maser.
In the case of a passive Rb vapor-cell clock it is helpful to evaluate the homogeneity of the H-field orientation across the cell volume. In order to optimize the fraction of the magnetic field component useful for driving the clock transition with respect to the total field energy over the cell volume, we define the field orientation factor (FOF) ξ as
ξ = VcellH 2 zdV Vcell|H| 2dV, (3)
where the z-axis is chosen as above. As for η, the maxi-mum value of ξ is 1 which implies the absence of the σ Zee-man transitions (H-field is exclusively z-oriented in this case). Note, that in contrast to η, the value of ξ does not depend on the ratio of the cell and the cavity volumes. Thus, ξ measures the proportion of the useful magnetic field (parallel to the C-field and the light) to the total C-field in the cell. For complete-ness, note that ξ is a good estimator of the DR signal qual-ity for buffer gas cells, where the alkali atoms can be treated as localized in space (on the timescales relevant for the DR signal).
The performances of the proposed magnetron cavity are reported in Table II. One can appreciate the high ξ value >0.85, while the filling factor (0.136) is more than two times less than the optimum value achievable in cylindrical resonators8(0.36) for the chosen cell dimensions.
TABLE II. Simulated microwave performance of the proposed magnetron cavity.
Parameter Symbol Value
Resonant frequency (GHz) fr 6.831
Filling factor η 0.136
Field orientation factor ξsim 0.877
Volume (dm3) V cavity 0.044 Q-factor (unloaded)a Q o 488 Q-factor (loaded)b Q l 185
aFrom eigenmode simulation. bExcitation, holes, and reservoir included.
As a matter of fact, η can be improved increasing the return flux cross-section, see Fig. 6, at the price of a larger shield radius.9Our design focused on the compactness of the structure, which provides an excellent working behavior be-cause of its high ξ as shown in Sec. V. The resonator cav-ity has a compact volume of <0.045 dm3 (with a cell of 0.012 dm3), in comparison to a TE011cylindrical cavity that would have a volume of0.14 dm3.
Finally, the loaded Q-factor (Ql<200) meets the passive atomic clocks requirements5 and ensures a negligible cavity pulling contribution on the atomic signal.28Qlis less than half the value of the unloaded Q-factor because of the presence of the apertures for the optical interaction.
B. Measured characteristics
The realized microwave cavity is illustrated in Fig. 7. The different pieces are made of surface treated aluminum and as-sembled using brass screws. A borosilicate glass cell contains Rb vapor and buffer gases.5 A 50 semi-rigid coaxial cable is used to obtain the desired loop excitation.
The microwave characterization was performed with a vector network analyzer (VNA) HP8720D. The cap and the
FIG. 7. Realized prototype (left). The tuning front cap attached to the cell and the coupling loop attached to the bottom enclosure are detailed on the right. 6.4 6.6 6.8 7 7.2 7.4 7.6 −25 −20 −15 −10 −5 0 Frequency [GHz] |S 11 | [dB] measurement sim.: tan δ =0 sim.: tan δ =0.05 6.4 6.6 6.8 7 7.2 7.4 7.6 −25 −20 −15 −10 −5 0 Frequency [GHz] |S 11 | [dB] measurement simulation (a) (b)
FIG. 8. Measured and simulated|S11| of the proposed magnetron cavity:
(a) glass cell filled with rubidium thermal vapor in the tuned configuration; (b) glass cell filled with air (for a slightly different tuning).
cell were screwed into the cavity to fine tune the resonator at 6.835 GHz. Two different effects are induced: first, the di-electric loading due to the glass cell reduces fr and, second,
the perturbation of the fringing field by the metal cap results in an upward fr shift. As the cap approaches the electrodes
(<4 mm), the latter effect becomes predominant.
Simulated and experimental performances are reported in Fig. 8(a). An agreement within 3% is noticed regarding the resonance frequencies, while a less close matching is ob-served in terms of the amplitude of the reflection coefficient |S11|. Experimental and numerical evidence suggests that this discrepancy may be due to the presence and the modeling of the Rb vapor and any remaining Rb in liquid phase. As re-ported in Fig. 8(b), the simulated and experimental results are almost superimposed when considering an empty cell (Rb as free space).
On the contrary, when considering the Rb vapor as lossy material (loss tangent tan δ= 0.05), numerical results match the measured performance at 6.835 GHz as shown in Fig. 8(a). However, further investigation is required with re-gard to this phenomenon.
Finally, two other resonating modes at ν = 7.17, 7.41 GHz are noticeable, which are not overlapping with the TE011-like mode at νRb(f≥ 5%).
FIG. 9. Clock experimental setup showing the laser head module used for stabilizing the laser, the physics package module containing the Rb cell inside a magnetron resonator, and the clock loop with the OCXO oscillator and microwave synthesizer. (BS) Beam splitter and (PD) photodetector.
IV. SPECTROSCOPIC EVALUATION OF THE MICROWAVE CAVITY
We have experimentally verified the microwave field dis-tribution inside the resonator by microwave-optical double-resonance spectroscopy in an atomic clock configuration. The potential of the resonator is demonstrated by measurement of the clock’s frequency stability in Sec. V.
A. Experimental setup
Figure 9 shows the schematics of our microwave-optical double-resonance experimental clock setup. A distributed-feedback (DFB) laser diode emitting at λopt = 780 nm acts as a source to optically pump the 87Rb atoms on the D2-transition (52S1/2→ 52P3/2).
An auxiliary 87Rb evacuated cell is used for saturated absorption spectroscopy in order to resolve the sub-Doppler peaks.29A first feed-back loop stabilizes the laser frequency to a selected sub-Doppler transition, derived from this aux-iliary cell. A compact laser head with a volume <0.87 dm3 was developed based on the above principle as explained in Ref. 29. For the laser head used in this study we mea-sured a linewidth <4.5 MHz, relative intensity noise (RIN) 7 × 10−14Hz−1and FM noise of 4 kHz/√Hz at 300 Hz.
The clock physics package (PP) module has an overall volume <0.8 dm3. It contains the clock cell with a diameter of 25 mm (filled with rubidium and buffer gases), mounted in-side our magnetron cavity (cf. Fig. 5) that is tuned to resonate around νRb. The cell volume is heated at a temperature Tv. A
dc magnetic field is applied in the direction of laser propaga-tion (along the z-axis) using coils that are wound around the cavity. This field isolates the87Rb ground-state clock tran-sition (52S1/2|Fg = 1, mF = 0 → |Fg = 2, mF = 0) by
lifting the degeneracy of the mF = 0 Zeeman sub-levels of
the ground-state hyperfine levels. The whole assembly is sur-rounded by a magnetic shield for suppressing fluctuations of the external magnetic field that otherwise would perturb the clock transition by second order Zeeman effects.
The microwave radiation injected into the cavity is produced by a synthesizer with a phase noise of <−108 dBrad2/Hz at≥300 Hz Fourier frequency (6.8 GHz carrier). For clock operation, a second feed-back loop established from the photodetector output to the microwave synthesizer is used to stabilize the microwave frequency to the87Rb clock transition.
B. Double resonance and Zeeman spectra
In the DR and clock experiments, the laser was stabilized to the Fg = 2 → Fe= 1,3 cross-over transition30 using the
auxiliary cell. The laser intensity incident to the clock cell was about 0.6 μW/mm2. The cell volume occupied by Rb va-por was regulated at Tv= 333 K. A microwave signal around
6.835 GHz with a power of−28 dBm was injected into the cavity. Under these optimized conditions, the DR signal has a narrow linewidth of 361 Hz and a contrast of 25%, see Fig. 10.
A wider span of the microwave frequency also shows DR signals due to higher Zeeman transitions with mF
= 0, as shown in Fig. 11. These σ -transitions are due to microwave magnetic field components perpendicular to the
FIG. 10. Double-resonance spectroscopic signal. Inset shows the de-modulated signal to which the microwave synthesizer is stabilized at the center.
FIG. 11. Zeeman transitions for an applied C-field of 44 mG. Suppression of σ-transitions is due to TE011-like H-field inside the cavity.
z-axis (considering the static field parallel to the z-axis) and are suppressed by∼97% compared to the central clock π-transition. The experimental field orientation factor ξexpis
determined using the equation, ξexp= Sπdν Sπdν+ Sσdν , (4) where Sπdν and
Sσdν are the transmission signal
strengths integrated with respect to the frequency detuning dν over all the corresponding Lorentzian peaks for mF= 0
(π -transitions) and mF = ±1 (σ-transitions), respectively.
For small microwave powers, the signal strengths Sπ are
pro-portional to H2
z, and Sσare proportional to Ht2(the transverse
field components).31The value ξ
exp= 0.868 computed from experimental spectra is in excellent agreement with the value estimated by simulations (ξsim= 0.877 according to Eq. (3)). This proves that the TE011-like cavity mode has a highly uniform magnetic field geometry oriented almost exclusively parallel to the laser propagating direction over the entire cell volume.
V. PERFORMANCE VALIDATION IN AN ATOMIC CLOCK
The short-term stability of a passive rubidium clock (ex-pressed in terms of Allan deviation, σy(τ )) can be predicted19
by the equation, σy(τ )= Npsd √ 2DνRbτ −1/2, (5)
TABLE III. Noise budget and estimation of signal-to-noise and shot-noise limits. Parameter Value FWHM 361 Hz Contrast 25% Discriminator 1.4 nA/Hz Npsd 3.2 pA/√Hz S/N limit 2.4× 10−13τ−1/2 Nshot 5.5× 10−14τ−1/2
FIG. 12. Short-term clock stability. Solid line indicates the measured stabil-ity, dotted line shows the signal-to-noise estimated value, dashed-dotted line shows the shot-noise limit, and dashed line shows the stability of an active H-Maser,32against which the clock is measured.
where Npsdis the total detection noise power spectral density (1 Hz bandwidth) measured when the microwave field and pump laser are switched on in a closed clock loop, νRbis the Rb ground-state hyperfine frequency, and D is the discrimina-tor slope of the error signal close to the line center (shown in the inset of Fig. 10).
Table III summarizes the noise budget for the short-term clock stability of our setup. For the clock stability’s shot-noise limit, the shot noise Nshotis calculated as
√ 2eIdc, where e is the electronic charge and Idc = 1.7 μA is the photocurrent of the DR signal at full width at half maximum (FWHM). For the short-term analysis of the clock stability we consider the integration time scale between 1 and 100 s.
For measurements of the clock stability, the 10 MHz output from the crystal oscillator (OCXO) (cf. Fig. 9) stabilized to the atomic clock transition is compared with a 10 MHz reference signal from an active H-Maser32 using a frequency comparator. The clock frequency is recorded using a computer interface. The measured clock stability of 2.4 × 10−13 τ−1/2 is in excellent agreement with the estimated signal-to-noise limit (see Fig. 12). This stability is approximately 2 times better than the previously reported result.19The increase is clearly due to the improvement of the electromagnetic field distribution of the proposed cavity (i.e., high field orientation factor). We would like also to point out that this result can still be improved towards the shot-noise limit by implementing a noise-cancellation method,33 or by using a less noisy laser source.
VI. CONCLUSION
A compact magnetron cavity for Rb frequency standards was presented. The design process started from the LRG lumped model and, showing the limitations, continued with a numerical optimization. The final prototype holds a glass cell of 25 mm and resonates with a uniform TE011-like mode at νRb. The simulated performances show a loaded Q-factor equal to 185, a filling factor η = 0.136, and a new field orien-tation factor ξsim = 0.877 with a volume <0.045 dm3. Such performances ensure an excellent working behavior for the targeted application.
Microwave characterization validated the simulated analysis albeit experimental results suggest the necessity of further investigation about modeling of the effect of the Rb. Measured performance (|S11(νRb)| = −6 dB) still allows clock analysis with a microwave power source as low as −28 dBm.
The suppression of σ -transitions in comparison with π-transition observed in the double-resonance experiment verified the high uniformity of the microwave magnetic field inside the cavity. This result confirms the necessity of the definition of the field orientation factor ξ (in addition to the well known filling factor) to properly characterize a resonator for Rb frequency standards. Excellent agreement is found between the simulated field orientation factor and experimentally evaluated FOF (ξsim = 0.877, ξexp = 0.868, respectively).
The achieved short-term clock stability of 2.4 × 10−13 τ−1/2using this compact microwave cavity demonstrates bet-ter stability than that of atomic clocks presently used in satel-lite navigation systems,17 such as the PHM (6.5 × 10−13 τ−1/2) and RAFS (2.5× 10−12τ−1/2). The realization of next generation portable atomic clocks using the demonstrated compact magnetron cavity for enlarged Rb cells of 25 mm diameter is thus of high interest for emerging demands in applications such as telecommunications, satellite navigation systems, local oscillators (LO) in portable optical frequency synthesizers and optical frequency standards.
ACKNOWLEDGMENTS
We acknowledge the support from the Swiss National Science Foundation (SNSF), the European Space Agency (ESA), and the Alliance and Swiss Space Office (SSO-SER). We also thank F. Gruet, P. Scherler, and M. Durrenberger from Unine-LTF, and C. E. Calosso from the Istituto Nazionale di Ricerca Metrologica (INRIM, Italy).
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